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Based on the stochastic maximum principle for the partially coupled forward-backward stochastic control system (FBSCS for short), a modified method of successive approximations (MSA for short) is established for stochastic recursive optimal control problems. The second-order adjoint processes are introduced in the augmented Hamiltonian minimization step since the control domain is not necessarily convex. Thanks to the theory of bounded mean oscillation martingales (BMO martingales for short), we give a delicate proof of the error estimate and then prove the convergence of the modified MSA algorithm. In a special case, we obtain a logarithmic convergence rate. When the control domain is convex and compact, a sufficient condition which makes the control returned from the MSA algorithm be a near-optimal control is given for a class of linear FBSCSs., Comment: 30 pages

In this paper, we propose a formulation to describe a risk-sensitive criterion involving asymmetric risk attitudes toward different risk sources. The introduced criterion can only be defined through the initial value of the minimal solutions of quadratic backward stochastic differential equations (BSDEs). Before uncovering the mean-variance representation for the introduced asymmetric risk-sensitive criterion by variational approach, some axioms to characterize a variance decomposition of square integrable random variables are given for the first time. The control problems under the asymmetric risk-sensitive criterion are characterized as a kind of stochastic recursive control problem that includes quadratic BSDEs. Under bounded and unbounded (linear-quadratic case) conditions, the asymmetric risk-sensitive control problems are investigated. In addition, a dynamic portfolio optimization problem featuring a stochastic return rate is provided as an application., Comment: 34 pages; All the authors share equal contribution to this paper

In this paper, we study a discrete-time stochastic optimal control problem under distribution uncertainty with convex control domain. By weak convergence method and Sion's minimax theorem, we obtain the variational inequality for cost functional under a reference probability $P^{\ast}$. Moreover, under the square integrability condition for noise and control, we establish the discrete-time stochastic maximum principle under $P^{\ast}$. Finally, we introduce a backward algorithm to calculate the reference probability $P^{\ast}$ and the optimal control $u^{\ast}$., Comment: 20 pages

Novel multi-step predictor-corrector numerical schemes have been derived for approximating decoupled forward-backward stochastic differential equations (FBSDEs). The stability and high order rate of convergence of the schemes are rigorously proved. We also present a sufficient and necessary condition for the stability of the schemes. Numerical experiments are given to illustrate the stability and convergence rates of the proposed methods.

In this paper, we study a stochastic recursive optimal control problem in which the value functional is defined by the solution of a backward stochastic differential equation (BSDE) under $\tilde{G}$-expectation. Under standard assumptions, we establish the comparison theorem for this kind of BSDE and give a novel and simple method to obtain the dynamic programming principle. Finally, we prove that the value function is the unique viscosity solution of a type of fully nonlinear HJB equation., Comment: 19 pages

We study a stochastic optimal control problem for forward-backward control systems with quadratic generators. The control domain is convex. By setting a bounded terminal value, we find that the variational equation for the backward state equation is a one-dimensional linear backward stochastic differential equation (BSDE for short) with unbounded stochastic Lipschitz coefficients involving bounded mean oscillation martingales (BMO-martingales for short), and the adjoint equation is a multi-dimensional BSDE with this type. Finally, a local stochastic maximum principle is obtained., Comment: 13 pages. arXiv admin note: text overlap with arXiv:2010.10680

A generalized Kalman-Bucy model under model uncertainty and a corresponding robust problem are studied in this paper. We find that this robust problem is equivalent to an estimate problem under a sublinear operator. By Girsanov transformation and the minimax theorem, we prove that this problem can be reformulated as a classical Kalman-Bucy filtering problem under a new probability measure. The equation which governs the optimal estimator is obtained. Moreover, the optimal estimator can be decomposed into the classical optimal estimator and a term related to model uncertainty., Comment: 10 pages

Recently, the deep learning method has been used for solving forward-backward stochastic differential equations (FBSDEs) and parabolic partial differential equations (PDEs). It has good accuracy and performance for high-dimensional problems. In this paper, we mainly solve fully coupled FBSDEs through deep learning and provide three algorithms. Several numerical results show remarkable performance especially for high-dimensional cases., Comment: 24 pages, 7 figures

In this paper, we study the solvability problem for one kind of fully coupled forward-backward stochastic difference equations (FBS{\Delta}Es). With the help of the necessary and sufficient condition for the solvability of the linear FBS{\Delta}Es, under the monotone assumption, we obtain the existence and uniqueness theorem for the general nonlinear ones., Comment: 19 pages. arXiv admin note: text overlap with arXiv:1812.11283

We study the Neyman-Pearson problem for convex expectations on L^{\infty}(\mu). The existence of the optimal test is given. Without assuming that the level sets of penalty functions are weakly compact, we prove that the optimal tests for convex expectations on L^{\infty}(\mu) are just the classical Neyman-Pearson tests between a fixed representative pair of simple hypotheses. Then we show that the Neyman-Pearson problem for convex expectations on L^{1}(\mu) can be solved similarly.

In this paper, we study the existence and uniqueness of viscosity solutions to a kind of Hamilton-Jacobi-Bellman (HJB) equations combined with algebra equations. This HJB equation is related to a stochastic optimal control problem for which the state equation is described by a fully coupled forward-backward stochastic differential equation. By extending Peng's backward semigroup approach to this problem, we obtain the dynamic programming principle and show that the value function is a viscosity solution to this HJB equation. As for the proof of the uniqueness of viscosity solution, the analysis method in Barles, Buckdahn and Pardoux Baeles-BP usually does not work for this fully coupled case. With the help of the uniqueness of the solution to FBSDEs, we propose a novel probabilistic approach to study the uniqueness of the solution to this HJB equation. We obtain that the value function is the minimum viscosity solution to this HJB equation. Especially, when the coefficients are independent of the control variable or the solution is smooth, the value function is the unique viscosity solution.

Hu et. al 2018 studied a stochastic optimal control problem for fully coupled forward-backward stochastic control systems with a nonempty control domain. By assuming a weakly coupled condition, they established an approach to obtain the first-order, second-order variational equations and the adjoint equations for the states X, Y and Z and deduced the global maximum principle. But it is well known that there are several different conditions such as monotonicity condition, weakly coupled condition and other conditions which can guarantee the existence and uniqueness of the solution to fully coupled FBSDEs. In this note, to overcome the limitations of assuming a specific condition, we propose two kinds of assumptions which can guarantee that the approach developed in Hu et. al 2018 is still applicable. Under these two kinds of assumptions, we obtain the global stochastic maximum principle., Comment: arXiv admin note: substantial text overlap with arXiv:1803.02109

Within the framework of viscosity solution, we study the relationship between the maximum principle (MP) in [9] and the dynamic programming principle (DPP) in [10] for a fully coupled forward-backward stochastic controlled system (FBSCS) with a nonconvex control domain. For a fully coupled FBSCS, both the corresponding MP and the corresponding Hamilton-Jacobi-Bellman (HJB) equation combine an algebra equation respectively. So this relationship becomes more complicated and almost no work involves this issue. With the help of a new decoupling technique, we obtain the desirable estimates for the fully coupled forward-backward variational equations and establish the relationship. Furthermore, for the smooth case, we discover the connection between the derivatives of the solution to the algebra equation and some terms in the first and second-order adjoint equations. Finally, we study the local case under the monotonicity conditions as in [14,27] and obtain the relationship between the MP in [27] and the DPP in [14].

This paper concerns the recursive utility maximization problem under partial information. We first transform our problem under partial information into the one under full information. When the generator of the recursive utility is concave, we adopt the variational formulation of the recursive utility which leads to a stochastic game problem and a characterization of the saddle point of the game is obtained. Then, we study the K-ignorance case and explicit saddle points of several examples are obtained. At last, when the generator of the recursive utility is smooth, we employ the terminal perturbation method to characterize the optimal terminal wealth., Comment: 20 pages

In this paper, the Neyman-Pearson lemma for general sublinear expectations is studied. We weaken the assumptions for sublinear expectations in [1] and give a completely new method to study this problem. Applying Mazur-Orlicz Theorem and the decomposition theorem of finitely additive set functions, we prove that the optimal test still has the reminiscent form as in the classical Neyman-Pearson lemma. Finally, for the special sublinear expectation which can be represented by a family of probability measures, we give a sufficient condition for the existence of the optimal test and show the form of the optimal test selected in L_{c}^1-space which is introduced by Peng [10] in his nonlinear-expectation framework., Comment: 15 pages

In this paper, we study the pricing of contingent claims under G-expectation. In order to accomodate volatility uncertainty, the price of the risky security is supposed to governed by a general linear stochastic differential equation (SDE) driven by G-Brownian motion. Utilizing the recently developed results of Backward SDE driven by G-Brownian motion, we obtain the superhedging and suberhedging prices of a given contingent claim. Explicit results in the Markovian case are also derived., 15 pages. arXiv admin note: substantial text overlap with arXiv:1212.5403, arXiv:1206.5889

In this paper, we study reflected backward stochastic difference equations (RBSDEs for short) with finitely many states in discrete time. The general existence and uniqueness result, as well as comparison theorems for the solutions, are established under mild assumptions. The connections between RBSDEs and optimal stopping problems are also given. Then we apply the obtained results to explore optimal stopping problems under $g$-expectation. Finally, we study the pricing of American contingent claims in our context., Comment: 29 pages

This paper is devoted to a stochastic differential game of functional forward-backward stochastic differential equation (FBSDE, for short). The associated upper and lower value functions of the stochastic differential game are defined by controlled functional backward stochastic differential equations (BSDEs, for short). Applying the Girsanov transformation method introduced by Buckdahn and Li [1], the upper and the lower value functions are shown to be deterministic. We also generalize the Hamilton-Jacobi-Bellman-Isaacs (HJBI, for short) equations to the path-dependent ones. By establishing the dynamic programming principal (DPP, for short), the upper and the lower value functions are shown to be the viscosity solutions of the corresponding upper and the lower path-dependent HJBI equations, respectively., Comment: We need to make a major change of this paper

In this paper, a stochastic optimal control problem is investigated in which the system is governed by a stochastic functional differential equation. In the framework of functional It\^o calculus, we build the dynamic programming principle and the related Path-dependent Hamilton-Jacobi-Bellman (HJB) equation. We prove that the value function is the viscosity solution of the Path-dependent HJB equation., Comment: We need to make a major change of this paper

In this paper, we study comparison theorem, nonlinear Feynman-Kac formula and Girsanov transformation of the BSDE driven by a G-Brownian motion.

In this paper, we study backward stochastic differential equations driven by a G-Brownian motion. The solution of such new type of BSDE is a triple (Y,Z,K) where K is a decreasing G-martingale. Under a Lipschitz condition for generator f and g in Y and Z. The existence and uniqueness of the solution (Y,Z,K) is proved. Although the methods used in the proof and the related estimates are quite different from the classical proof for BSDEs, stochastic calculus in G-framework plays a central role., Comment: 27 pages

This paper formulates a model of utility for a continuous time framework that captures the decision-maker's concern with ambiguity about both the drift and volatility of the driving process. At a technical level, the analysis requires a significant departure from existing continuous time modeling because it cannot be done within a probability space framework. This is because ambiguity about volatility leads invariably to a set of nonequivalent priors, that is, to priors that disagree about which scenarios are possible., Comment: 39 pages

We give a theory of sublinear expectations and martingales in discrete time. Without assuming the existence of a dominating probability measure, we derive the extensions of classical results on uniform integrability, optional stopping of martingales, and martingale convergence. We also give a theory of BSDEs in the context of sublinear expectations and a finite-state space, including general existence and comparison results.

We obtain an existence and uniqueness result for an initial coupled forward-backward stochastic differential equation under some monotone conditions. Then this result was applied to backward linear-quadratic nonzero-sum stochastic differential game problem and get the explicit form of a Nash equilibrium point.